A059533 - OEIS

%I #26 Jul 20 2024 12:49:43

%S 1,3,5,7,9,11,13,15,17,19,21,22,24,26,28,30,32,34,36,38,40,42,44,45,

%T 47,49,51,53,55,57,59,61,63,65,67,68,70,72,74,76,78,80,82,84,86,88,90,

%U 91,93,95,97,99,101,103,105,107,109,111,113,114,116,118,120,122,124

%N Beatty sequence for 1 + Catalan's constant.

%H Harry J. Smith, <a href="/A059533/b059533.txt">Table of n, a(n) for n = 1..2000</a>

%H Aviezri S. Fraenkel, Jonathan Levitt, and Michael Shimshoni, <a href="http://dx.doi.org/10.1016/0012-365X(72)90012-X">Characterization of the set of values f(n)=[n alpha], n=1,2,...</a>, Discrete Math. 2 (1972), no.4, 335-345.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BeattySequence.html">Beatty Sequence</a>

%H <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>

%F a(n) = floor(n*(1+A006752)). - _R. J. Mathar_, May 22 2019

%t Floor[Range[100]*(1 + Catalan)] (* _Paolo Xausa_, Jul 05 2024 *)

%o (PARI) { numdigits=100; default(realprecision, numdigits+80); s=1.0; n=5*numdigits; j=4*n+1; si=-1.0; for (i=3, j-2, s+=si/i^2; si=-si; i++; ); s+=0.5/j^2; ttk=4.0; d=4.0*j^3; xk=2.0; xkp=xk; for (k=2, 100000000, term=(ttk-1)*ttk*xkp; xk++; xkp*=xk; if (k>2, term*=xk; xk++; xkp*=xk; ); term*=bernreal(k)/d; sn=s+term; if (sn==s, break); s=sn; ttk*=4.0; d*=(k+1)*(k+2)*j^2; k++; ); b=1 + s; for (n = 1, 2000, write("b059533.txt", n, " ", floor(n*b)); ) } \\ _Harry J. Smith_, Jun 27 2009

%Y Beatty complement is A059534.

%K nonn,easy

%O 1,2

%A _Mitch Harris_, Jan 22 2001